First of all apologies is this problem comes up ad nauseam; I have done a search on the forums but have come up with nothing that could help solve my problem. Might just be due to my incompetence.
Anyway, I have the problem mentioned in the topic title: my rear derailleur shifts very well up to top gear (7th), but when I shift down to 6, the chain stays on the same sprocket. Going down to 5th, 4th, 3rd and 1st will move the chain one gear, accordingly - meaning I can only go down to the second largest sprocket.
Request: HELP!
Might also be worth something:
When in 7th, the derailleur makes some sort of "catching" noise. When I shift down to 6 the noise stops.
... should I adjust the "H" screw on the derailleur? Or does my problem come from elsewhere?
Thank you in advance for your replies! I'll be keeping away from 7th gear until I get this sorted.
Dan

Try turning the barrel adjuster on the derailleur clockwise a 1/4 of turn at a time. I have had the same problem with the "catching" noise and have found it to be when ever you click your shifter into your high gear the derailleur doesnt move quite far enough therefore not shifting all the way into its highest gear.
I have learned only by personal experience, so if anyone wants to back me up or correct me please feel free.

Yes I have had the same problem. It would shift fine from 1 thru 7 but would not quite shift far enough to 8th(24speed). Turning the barrel adjuster clockwise should alleviate the problem. I think turning the H or L screws might make it worse. But like I said Ive only learned from triala and error, so you may want to get a second opinion from a more seasoned person on here, like say Alex or Cycler UK, them 2 have helped me out a lot.

Thank you for your replies so far, Stan (hope I didn't get the name wrong).
I hope I didn't create any confusion when I mentioned 7th gear. That is actually how many sprockets I have - 7-speed Shimano Acera package. So shifting all the way up is fine. Shifting down is not.
Any further assistance will be rewarded with the beverage of your choice whenever you happen to be in Romania.

If shifting down is your problem and the h screw is set correctly then the likely cause is friction between your inner and outer derailleur cable housings  or ferrules Is the cable slack when you try to shift to your highest gear?
Moreover, is your problem you cannot thereafter get first after selecting seventh?

That's exactly my problem, Marc. After selecting seventh, I can't go down to first.
Any suggestions on how I could fix this? I should mention that my mechanical skills are limited when bicycles are concerned.

right, this should be quite simple. your mech has a spring in it. it also has a cable which is attached to the shifter on the handlebar. in its natural state the spring will push the mech onto the smallest spocket which i assume is gear number 7,ok. then as you click the shifte to 6, 5, 4, etc you pull the cable against the spring and move the mech up into a larger sprocket. in your case the cable is probably a bit loose. put it in 7 and feel the cable coming fom the mech. if it is a bit slack then you have found your problem, well done. you must now tighten the cable. DO NOT ADJUST the 'H' screw unless you want the chain coming off the end of the sprockets, all right ? . To tighten the cable, put it in 6 gradually turn the barrel adjuster on the mech anti clockwise while slowly turning the cranks the chain should click a bit and then jump up to 6 where it should be, nice.
There are other ways to tighten the cable if this doesnt work but I cant be bothered to type any more so let us know,

I would tend to agree with Marcdominic above.
With this type of fault, hand pressure easily overcomes any stiction when going "up" through the gears (small to large cogs). Whereas when going large to small cogs you are relying on the derailleur return spring to force it back. So any cable drag / resistance shows up by "slow" selection of the next smaller cog.
You've not said what kind of bike (road or MTB) or what your make of gears are. (STI etc)?
If it's a normal road bike, and the cable goes under the bottom bracket, then make sure this area is clean.
All the dirt thrown from the front wheel ends up here.
There is a plastic piece with 2 groves that the cables run through and can clog up.
This causes the inner cable to drag when going from large to small cogs.
Run up through the gears and put the chain on the big cog.
Now just flick the gear changer, without moving the back wheel, back to a smaller cog. (That should leave the chain on a large sprocket.)
This will slacken the cable and allow you to check if it slides through the guide under the BB.
If this isn't the fault then remove the inner and look for loose wire strands or kinks.
Clean it and replace if it looks O.K.
Check the outer isn't kinked or has tight bends.
Also, if you can, clean the derailleur to make sure that it moves freely.

Hand pressure only overcomes stiction going 'up' the block if the gear cable is properly tensioned. If the cable is loose you will never get all the way up the block no matter how new the cable is Tightening it slightly as descibed above is always the first thing to try. This should also give you first gear ( largest sprocket). Ok?

Now I know technically top gear is the small cog but people tend to call gears by the number on the changer.
We need to know what type of gears Dan has got.
No.1 gear on all my bikes is the small cog. (High Normal).
(Yes some cycles now have the No.1 = largest sprocket).
Dan also says it will go to "top" gear (7th) so the assumption is the cable isn't slack?
If the cable was slack then it would go to the smallest cog with no problem.
He also says it chatters in 7th but not drop back to 6th. So if anything the cable could be to tight.

Listen, dont know what you mean. Ive been around the block a few times believe me and I know a thing o two about bikes. if we are talking about a 7 speed ace ra mech then the smallest sprocket will be 7 on the shifter, thats what he means, allright yeah. Im certain ,100% certain in fact. So forget about a tight cable ok. Just do what i've said in my earlier post and we'll take it from there.
FYI- the vast majority of drivetrains are set up like this
sorry to be blunt but you gotta be youself in this life and im just trying to help people out ....

Alright, so I took chainbrain's advice because it seemed to make sense at the time. Shifted to 7th gear (top gear; last gear; smallest sprocket, what have you...) and the cable had quite a bit of slack; so I changed down (up!?) to 6th and adjusted the barrel adjuster till the chain got to where it's supposed to be. Now I can shift fine up to 7th and back into 6th.
Only now it won't go down to 5th! And the catching sound is still there. I'll try to make a small video tomorrow when I get back home from work, so you can see the symptoms for yourself.
To answer all the questions I've seen so far, the bike is a hardtail mountain bike; 19" Al 7005 frame with Shimano Acera rear derailleur and cassette sprocket (1st gear being the largest cog) and Shimano Tourney at the front; Acera shifters and that's all I consider relevant to this topic. If you feel a strong urge to, please don't comment on the low quality of the components. Come autumn, the bike is due for an upgrade to Deore XT Shadow and Rock Shox fork.

you should be able to get a balance by rotating the barrel adjuster clockwise again ,ok. It helps to have the rear wheel off the ground so you can turn the cranks and fiddle with the adjuster .. As you probably don't have a workstand you can have a friend lift it up for you. If it still wont index properly replace the cable -and the outer if it isn't mint. If you don't want to get a new cable then spray a load of wd 40 into the cable outers (wipe off the excess lube) which might work but i doubt it..

Chainbrain, I think you are exactly right. A new cable will usually solve these kind of shifting problems. Slick Honey is the only cable lubricant I've ever come across that will get an old cable working, but a new inner and outer cable is still better (Slick Honey on the new cable will work a treat too).
The other possibility is that the derailleur hanger is bent. The derailleur needs to pivot in a plane that is parallel with the plane of the rear wheel. If you have had a fall on the right side of the bike, there is every chance the derailleur hanger is bent inwards. You can see this if you look at the bike from behind and the derailleur cage is angled inwards ( ...scrape marks on the derailleur is a give-away also...) If so, you need a hanger aligning tool or it's time to visit a good bike shop to get the job done there.

ftpaddict,
If after using seventh gear and then you cannot select first (all on rear cassette), how do you manage to then get it back into first to be able to demonstrate your observed symptoms? Do you have to manually adjust the rear derailleur (by hand to get this transmission?
Furthermore (I think you have multiple chainrings up front), does this behaviour exhibit itself in all gear combinations (Im not suggesting you should actually ride in all combinations but for the purposes of diagnostics with the bike off the ground this can be done)?
Let us know

chainbrain,
Just re-read the thread: to clarify by seventh I mean the smallest cog on the rear cassette  first would be the largest I call em the same way you do. Of course seventh could be seventh, fourteenth or twenty-first depending on your chainset setting
Even more confusingly, by shifting down I meant shifting up (i.e. from first to seventh which is down on the rear cassette): Ill try to be more consistent in the future  apologies if I miss-lead
I think we need additional information from ftpaddict in this case. The problem certainly sounds intriguing

Alright, everyone. I apologize for not saying anything for so long. All my time has been taken up with work and family matters. With that out of the way, I can now tell you that I have the problem solved. It wasn't the cable's fault; it was mine. I needed to tension it more. I was afraid of overdoing it, so I didn't do it properly. Last Saturday I went to scout for a new fork, and the store I got to had a service area (Trek only, but the guys there were so helpful they fixed it all up in 5 seconds and they didn't even ask for anything in return - I'm definitely buying my fork from them, even if it's a little bit more expensive).
Now there's just that small matter of adjusting the L screw, because I fiddled with it before asking for help here, and I'm pretty sure that's what's causing my inability to shift into 1st. All the other gears work very well.
chainbrain, I appreciate your interest in this problem, but there is no need to be over-zealous. Everybody just wants to help.
The bike is almost new; I should think there is no need for a new cable just yet. (:
Thank you one and all for your insight and suggestions. I hope I won't be having any other issues with my bike, but if I do, this will be the first place I stop by to ask a question.
Dan